A Carleman type theorem for proper holomorphic embeddings

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Ark. Mat., 35 (1997), 157-169

A Carleman type theorem for proper holomorphic embeddings

Gregery T. Buzzard and Franc Forstneric

1. I n t r o d u c t i o n

We denote by C the field of complex numbers and by R the field of real numbers. To motivate our main result we recall the C a r l e m a n approximation t h e o r e m [4], [11]: For each continuous function /~: R - - ~ C and positive continuous function

~1: R--*(0, c~) there exists an entire function f on C such that If(t)-)~(t)l<~l(t) for all t E R . If ~ is smooth, we can also a p p r o x i m a t e its derivatives by those of f . A more general result was proved by Arakelian [2] (see [14] for a simple proof).

Let C ~ be the complex Euclidean space of dimension n. Our main result is an extension of C a r l e m a n ' s t h e o r e m to proper holomorphic embeddings of C into C n for n > l :

1.1. T h e o r e m . Let n > l and r>_O be integers. Given a proper embedding A: Rr '~ of class C r and a continuous positive function 7: R--*(0, oc), there exists a proper holomorphic embedding f: C~-~C n such that

If(s)(t)-)~(~)(t)i<~(t), t E R , 0 < s < r .

If in addition T = { t j } c R is discrete, there exists f as above such that f ( s ) ( t ) = A (s)(t), t C T , 0 < s < r .

Definition. Two proper holomorphic embeddings f,g: Cr n are said to be Ant Ca-equivalent if Oof=g for some holomorphic a u t o m o r p h i s m 9 of C n.

1.2. C o r o l l a r y . For each n > l the set of A n t Ca-equivalence classes of proper holomorphic embeddings Cr n is uncountable.

For n > 3 the corollary is due to Rosay and Rudin [16]. T h e corollary follows from T h e o r e m 1.1 and a result of Rosay and Rudin [15] to the effect t h a t for each

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n > l there exist uncountably m a n y discrete sets in C n which are pairwise inequivalent under the group of holomorphic a u t o m o r p h i s m s of C ~. T h e o r e m 1.1 provides for each discrete set E = { e k : k = l , 2,3, . . . } c C n a proper holomorphic embedding fE: C r such t h a t f E ( k ) = e k for all k = l , 2, 3, .... (For n > 3 such embeddings were constructed in [16].) Clearly the embeddings corresponding to inequivalent discrete sets are inequivalent.

In this context we recall t h a t the first construction of proper holomorphic embeddings Cr 2 which are inequivalent to the s t a n d a r d embedding ~ - * (r 0) by a u t o m o r p h i s m s of C 2 can be found in [8]. On t h e other hand, it is well known t h a t all polynomial embeddings Cr 2 are equivalent to the s t a n d a r d embedding by polynomial a u t o m o r p h i s m s of C 2 [1], [18].

Remark 1. We emphasize that, in T h e o r e m 1.1, one cannot expect in general to extend ~ to a holomorphic embedding of C into C ~. If A is real-analytic, it will extend holomorphically to some open set in C, b u t in general not to all of C;

and even if A extends to all of C, the (unique!) extension need not be a proper m a p into C% So the best we can do in general is to a p p r o x i m a t e ), by a proper holomorphic embedding as in T h e o r e m 1.1.

Remark 2. If the embedding A: R~--*C ~ is of class g ~ , our m e t h o d can be modified so t h a t we a p p r o x i m a t e to increasingly high order on complements of c o m p a c t subsets of R. Another possible extension is to a p p r o x i m a t e a proper smooth embedding by a proper holomorphic embedding on a set of disjoint lines or other real curves in C. We shall not go into details of this.

T h e original motivation for T h e o r e m 1.1 was the question, communicated to us by R. Narasimhan, as to whether there exist proper holomorphic embeddings f: C ' ~ C 2 such t h a t f ( C ) is a nontrivial knot in C 2, i.e., C 2 \ f ( C ) is not homeomorphic to C 2 \ ( C x {0}), the complement of the embedding ~-+(~, 0). Unfortunately we have not been able to construct such embeddings with the aid of T h e o r e m 1.1 because real one-dimensional curves in C 2 ~ R 4 are always unknotted.

In order to place T h e o r e m 1.1 in context we recall some recent developments on embedding Stein manifolds in C n from [3], [7], [8], [9]. (For Stein manifolds and other topics in several complex variables mentioned here we refer the reader to H h r m a n d e r [12].) In those papers it was shown t h a t a Stein manifold M which admits a proper holomorphic embedding in C n for some n > l also admits an embedding f:Mr n whose image f ( M ) c C n contains a given discrete subset E c C ~ [7, T h e o r e m 5.1]. (Recall t h a t any Stein manifold M embeds in C n for n > 8 9 according to Eliashberg and G r o m o v [5].) W i t h m e t h o d s of the present p a p e r one can show moreover t h a t for each pair of discrete sets A =

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A Carleman type theorem for proper holomorphic embeddings 159

{ a j } ~ _ l c M and E = { e j } ~ _ l C C n there exists a proper holomorphic embedding f : M~-+C n such t h a t f ( a j ) = e j for j = l , 2,3, ....

In light of this, a n a t u r a l question is whether one can replace discrete sets in M by certain positive dimensional submanifolds N c M , i.e., when is it possible to a p p r o x i m a t e a s m o o t h proper embedding )~:N---+C n by the restrictions to N of proper holomorphic embeddings f : Mr For compact, totally real, holomorphically convex submanifolds N c M the answer is affirmative and it follows immediately from t h e approximation theorems in [61 a n d [10]. T h e case when N is noncompact is much harder. Our m a i n result in this p a p e r provides an affirmative answer in the simplest such case when M = C and N = R • { i 0 } c C . It seems likely t h a t the result remains valid when N is a properly e m b e d d e d real line in any Stein manifold M. T h e details of our construction are considerable, even in this simplest case, and the full scope of the m e t h o d remains to be seen.

T h e p a p e r is organized as follows. In Section 2 we introduce the notation and give an outline of t h e proof of T h e o r e m 1.1. In Section 3 we collect some technicM lemmas needed in the proof. T h e details of the proof of T h e o r e m 1.1 are given in Section 4.

T h e second author acknowledges partial s u p p o r t by an NSF grant and by a grant from the Ministry of Science of the Republic of Slovenia.

2. O u t l i n e o f p r o o f

Since the proof of T h e o r e m 1.1 is somewhat intricate, we give in this section an outline of the proof. We also recall a technical result from [10] (Proposition 2.1 below) which will be used in the proof.

We begin by explaining the notation. We denote b y A~ the closed disc in C of radius O and center 0, by B the open unit ball in C n with center 0, and by R B the ball of radius R. For a set A c C n and p>O, let A + o B = { a + z : a E A , [zl<_O}. We identify C and R with their images in C n under the embedding @-+(~,0, ... ,0).

For l<_j<_n we denote by ~j the coordinate projection Irj(zl ,... , z~)=zj.

In the proof we shall use special a u t o m o r p h i s m s of C n of the form 9 (z) =z+f(~rz)v, z 9 C ~,

where v 9 ~, 7r: C~--+C k is a linear m a p for some k < n with 7rv--O (in most cases k = l ) , and f is an entire function on C k. An a u t o m o r p h i s m of this form is called a shear; clearly ~ - 1 (z) = z - f(Trz)v.

One of the m a i n technical tools in our construction is the following result from [10]. The case r=O was obtained earlier in [9]. This result can also be obtained by methods in [6].

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2.1. P r o p o s i t i o n . Let K c c C n (n_>2) be a compact, polynomially convex set, and let C c C n be a smooth embedded arc of class C ~ which is attached to K in a single point of K . Given a C a diffeomorphism F: K U C - + K U C ' c C n such that F is the identity on ( K U C ) N U for some open neighborhood U of K , and given numbers r>_O, r there exist a neighborhood W of K and an automorphism (I)EAut C n satisfying

II~-Idllc,-(w)<~, II~-FIIc,-(c) <c.

(Here Id denotes the identity map.) Moreover, for each finite subset Z C K U C we can choose q) such that it agrees with the identity to order r at each point z E Z N K and q~[c agrees with F to order r at each point of Z N C .

T h e same result holds with any finite n u m b e r of disjoint arcs attached to K . We now give the outline of the proof of T h e o r e m 1.1. We wish to a p p r o x i m a t e a given proper embedding A: R~-~C n by the restriction to R of a proper holomorphic embedding f : C~-~C n. By s t a n d a r d results we m a y assume t h a t A is Coo and t h a t any C r m a p of R into C n which has C r distance less t h a n r/(t ) from/~ (as in T h e o r e m 1.1) is a proper embedding.

We start with the standard embedding ? 0 ( t ) = ( t , 0,... , 0) and identify R with

"y0(R). We inductively define a u t o m o r p h i s m s of C n of the form fk=q~k ... iPl o91 o .... ~k, where each (I)j and

II/j

is an a u t o m o r p h i s m of C n chosen so t h a t fkl~o(n) approximates A on larger and larger compact sets. Moreover, we construct the sequence fk such t h a t the limit f=limk-+oo fk exists on an open set D c C n containing C x {0}, and f is a biholomorphic m a p of D onto C ~. The restriction of f to the Zl-COordinate axis is then a proper holomorphic embedding of C = C x {0} into C n satisfying the required properties.

T h e inductive correction proceeds as follows. We assume t h a t there is an interval Ik c R such t h a t fk approximates/~ on Ik in the sense of T h e o r e m 1.1 and t h a t b o t h A(t) and fk(t) lie outside some closed ball Bk for t ~ I k . We want to produce an interval Ik+l and a ball Bk+l, each of which has radius at least one greater t h a n the corresponding set at the kth stage, and a m a p fk+l which gives the desired approximation on Ik+l.

We do this by applying a version of Proposition 2.1 to get an a u t o m o r p h i s m

~ k + l which is close to the identity m a p on the sets fk(Ik), Bk, and f k ( A k + l ) , and such t h a t ~k+lofk approximates A on Ik+l. In order to apply Proposition 2.1, we define a polynomially convex set Kk which includes Bk, f k ( A k + l ) , and most of fk(Ik), and we also define a smooth, proper embedding Ak: R - - + C ~ which agrees with )~ on R \ I k , agrees with fk. on most of Ik, and is C~-near )~ everywhere. We can then apply Proposition 2.1 via L e m m a 3.4 to get r so t h a t (I)k+l o fk approximates ,~ on some larger interval Ik+l and is near the identity on Kk (this is required for

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convergence).

The problem now is that the point ~ k + l o f k ( t ) may come very close to the previous ball Bk for some t E R \ I k + l . Unless we control this distance from below, the limit map m a y not be a proper embedding. Hence we precompose Ok+l ~ fk with a shear of the form

~k+l(Z)=Z-}-Pk+l(Zl)Pk+l,

for some #k+l holomorphic in one variable and some vector ~k+l with 7rl~k+1----0. By L e m m a 3.2, we can choose ~ k + l near the identity on Ak+l UIk+l and such that ~Pk+l~176 ( R \ I k + l ) avoids some larger ball Bk+l. Except for technicalities, this finishes the induction.

T h e proof is completed by showing that f = l i m k ~ fk exists and gives a proper holomorphic embedding of C = C • {0} to C n. This is so because the limit 9 =limk--+~ ~1 . . . ~k exists and is an automorphism of C n, while the limit ~---- l i m k ~ ~k ... ~1 exists on an open set ~ c C ~ containing q2(C• and ~P: gt--+

C n is a biholomorphic map onto C ~. Thus f = ( I ) o ~ is a biholomorphic map from D = ~ -1 (~) onto C n whose restriction to C • { 0 } c D provides the desired proper holomorphic embedding into C ~. The approximation properties of f are clear from the inductive step.

3. S o m e l e m m a s

The following is standard, e.g., [13, Proposition 2.15.4].

3.1. L e m m a . Let A: Rr n be a C ~ proper embedding. Then there exists a continuous ~: R--+ (0, oc) such that if V: R--+ C n with IV (s) (t) -)~(s) (t) l < ~(t) for all tEl:t, s = 0 , 1, then 7 is a proper embedding.

Recall t h a t a compact set A C C C n is polynomially convex if for each z c C ' ~ \ A there is a holomorphic polynomial P on C n such that IP(z) l > m a x { I P ( w ) l : w e A } . We refer the reader to [12] for properties of such sets.

3.2. L e m m a . Let A c C n be compact and polynomially convex and Q>0. Let I C R be an interval whose endpoints lie in c n \ ( A U / k e ) , and let r , c > 0 . Then there exists an automorphism ~ ( z ) = z + g ( z l ) e 2 of C ~ such that

(i) IV(z)-zl< for

(ii) IIq21R(t)-tllc~(i ) <c, and (iii) ~ ( t ) ~ A for t e R \ I .

If Z C I is finite, we can choose 9 as above so that g(S)(t)=0 for t E Z , 0 < s < r . Proof. Let #1 <#2 denote the endpoints of I in R , and let F j - - { ( # j , 4, 0 , . . . , 0):

~CC} for j = l , 2. Let R > m a x { I t t l l , 1#21}+1 such that A c R B . Consider the set E j = A A F j . Since A is polynomially convex, E j is polynomially convex in Fj and hence F j \ E j is connected. Since the endpoints of I lie in F j \ E j , there exists a

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smooth curve 3'j: [0, 1 1 - ~ F j \ E j with 7 j ( 0 ) = ( # j , 0,... , 0) and [Tr27j(1)[ > R + I for j = 1 , 2 .

Since A is compact, there exists 6 > 0 such that ~/j([0, 1 ] ) + 3 ~ B c C n \ A . Let 7r2(z)=z2. Let K = { x + i y E C : # l - ~ ( 5 < x < # ~ + ~(5, lY[ 1 1 < ~ + 1 } . Define a function h: K U [ - R , R ] - + C by

/ ~r271 (1), if t E [ - R , #1 - 2~1;

712~/1 ( ( ~ l - 6 - t ) / 6 ) , if t E [#1--2~, #1--~];

h(t) = 7r2~/2((t-1~2-6)/6), if t E [/z2+6,#2+2~];

7r272 (1), if t E [p2 +26, R];

0, otherwise.

Choose r/, 0 < r / < m i n { e , 6}. By Mergelyan's theorem [17, Theorem 20.5] there is an entire function g on C such that I h ( z ) - g ( z ) l < n for z E K U [ - R , R]. The shear 9 (z)=z+g(zi)e2 then satisfies (i) and (iii). Since I C I n t K , Cauchy's estimates imply that it also satisfies (ii) provided that r]>0 is chosen sufficiently small. The last condition on g is a trivial addition to Mergelyan's theorem. []

3.3. L e m m a . Let A:Rr n be a proper, C ~ embedding, ^{K c C} ⁿ compact, e > 0 , and r E Z + . Let Z c R be finite, and suppose A ( t ) E C = C x { 0 } n-~ for each t E Z . Then there exists a shear q~(z)=z+h(zl)v for some v E C n with 7rlV=O such that

(i) ~P(C) M)~(R) =)~(Z),

(ii) I O ( z ) - z l < e for z E K , and

(iii) qJ(z)=z+O(]z-A(t)] r+l) as z--+A(t), for all t E Z .

Proof. Let Z={ty}~=l. For CEC let h(~)=Hl<_j<_~(r ~+1. Consider the map @: CxC'~-I---~C ~ given by

e ( z l , , . . . , = (zx, 0 , . . . , 0 ) + h ( z l ) ( 0 , , . . . ,

Clearly q5 is an automorphism of (C\A(Z)) x C n-1. Let An,j denote the closed disc of radius R in C with center ~rlA(tj) for j = l , 2,... , s. Choose R > 0 such that the discs An,j for l<_j<s are pairwise disjoint. Choose a p, 0 < t ) < R , such that t) 2 is a regular value of #j(t)=l~rlA(t)-TrlA(tj)] 2 ( t E R ) for each j = l , 2,... ,s.

Let Mo=C\Ul<_j<_sintAo, j. Let ~o=~[MoxC~-i and O c ~ = - f f 2 l o M o x C n 1. A simple check shows that B e and 0qse are transverse to A(R). Hence by the transversality theorem, there exists a set A o c C n-1 of full measure such that for each c~=(c~2, ... , a n ) E A o, ~ ( M o, c~)={~(Zl, c~):Zl EMe} and A(R) are transverse, hence disjoint by dimension considerations.

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A Carleman type theorem for proper holomorphic embeddings 163 A

Let

=Nj=IA1/j.

Then

A c C ~-1

has full measure, and for each a E A we see that

O(C\A(Z), (~)

and )~(R) are disjoint. Finally, choose

a e A

such that ]h(zl)c~] < e for z l e ~ ( g ) , and let

~(z)---z+h(Zl)a.

T h e n ~ ( z l , 0 , . . . , 0 ) = ~ ( z l , a 2 , . . . , a n ) , and 9 satisfies the conclusions of the lemma. []

3.4. L e m m a .

Let/~:

Rr n

be a C a embedding, f:

Cr n

a proper holomor- phic embedding, and I c R a closed interval with f]i---A[r. Let K c C n be compact and polynomially convex, a,

r , s > 0 ,

and

T C R

discrete. Suppose that A ( t ) , f ( t ) ~ K for t E R \ I . Then there exists

~ E A u t C '~

such that if g={~of, then

(i)

]g(S)(t)-A(s)(t)[<r for te[-a,a], O<_s<_r,

(ii)

g(~)(t)=i~(~)(t) for t e T n [ - a , a ] , O<_s<_r, and

(iii) ] O ( z ) - z ] < s

for z e K .

Proof.

We may assume that

I c ( - a , a ) .

Let

I1, I2

be the two connected components of

{ ~ E I : f ( ~ ) E C n \ K }

containing the respective endpoints of I, and let

Io=I\(IiUI2).

Let A be the polynomial hull of

KUf(Io).

T h e n A is the union of

KUf(Io)

and the bounded connected components of

f ( C ) \ ( K U f ( I o ) ) .

Note t h a t f ( I 1 ) and f ( I 2 ) lie in f ( C ) \ A since

f ( t ) ~ K

for all

t e R \ I .

Let

L = A U I ( [ - a ,

a]). T h e n

C = L \ A

is the union of two embedded arcs, each containing an endpoint of

f([-a,

a]). Define F on L by

F ( z ) = z

if

z e A ,

and F(z)--- )~f-1 (z) if

z E f ( [ - a ,

a]). T h e n F is a C ~ diffeomorphism of L which extends as the identity map on

(AUC)NU

for some neighborhood U of A. Apply Proposition 2.1 to get O E A u t C ~ such that ] r for

z e K

and such that

g=(~of

satisfies (i) and (ii). []

4. P r o o f o f T h e o r e m 1.1

Choose a smooth cutoff function X on R such that x ( t ) - - 1 for It] small and s u p p x C ( - 1 , 1). Define the constant

C = C r > I

such t h a t

IIxh][c~<C[[h[[c ~

for each h e g r ( R ) . We fix such C for the entire proof.

By approximation we may assume t h a t )~: Rr n in T h e o r e m 1.1 is a proper C ~ embedding. Decreasing ~] if necessary we may also assume that ~? satisfies L e m m a 3.1 for A and ~ ( t ) < 1 for all t E R .

We use an inductive procedure to obtain a sequence of proper holomorphic embeddings

fk: C~-+C n

such that f = l i m k ~ fk exists on C and satisfies T h e o r e m 1.1.

Each fk will be a restriction to

C = C • (0} n-1

of a holomorphic automorphism of C n. T h e next map

fk+l

will be of the form

fk+l=q~k+]ofko~k+l

for suitably chosen ~k+l, ~k+l EAut C n.

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We will describe the case k = 1 after the inductive step is given. Recall that Ak is the closed disc in C = C x {0} n-1 with center 0 and radius k, and B is the unit ball in C n. For the induction at step k, suppose we have the following:

(a) closed balls Bj =

Rj

B c C ~ with

Rj ~

m a x { j + 1, R j _ 1 ~- 1}, j = 1 , . . . , k, (b) automorphisms O1 ,... ,Ok of C a with

IOj(z)-zl<2-J

for

zEBy_I, j=

2,... ,k,

(c) numbers ~ j > 0 such that c1<2 -1 and

c j < l c j _ l < 2 -j

for

j=2, ... ,k,

(d) automorphisms ~ 1 , . - . , ~ k of C a of the form

q2j(z)=z+gj(zl)e2+

hj(zl)vy,

where 7 0 ( v j ) = 0 and I ~ j ( z ) - z ] < s j for

Izl<_j,

(e) closed intervals

Ij = [-aj, aj], j = 1,... , k,

with aj > m a x { a j _ l

+2, j +2},

and

(f) numbers 0<by < 6 -1

inf{v(t):te/j}, j = l , . . . , k,

such that the automorphism

fk = Ok .... O1o~1 . . . ~k E Aut C n

(whose restriction to C = C • {0} n-1 provides an embedding Cr n) satisfies:

(lk) f k ( A j + c k B ) C I n t

By

for j = l , . . . , k, (2k)

If(~)(t)-;~(s)(t)l<~(t)

for

telk

and 0 < s < r ,

(3k)

If(~)(t)-)~(s)(t)l<hk

for

tEIk\(--ak+l, ak--1)

and 0 < s < r , (4k)

f(~)(t)=)~(~)(t)

for

tETNIk, O<_s<_r,

(hk)

fk(C)N;~(R)=A(TNIk),

(6k) I)~(t)l>Rk+l for

Itl>_ak-1, (7k) IA(t)l>Rk for Itl>ak-1.

We will now show how to obtain these hypotheses at step k + 1. Let I 1 and I~ be the two connected components of the set {~ C Ik \Ak+1:1 fk (~)1 > Rk } containing the respective endpoints of the interval

Ik.

Let 0 _

I~-Ik\(I~ UIk)

1 2 be the middle interval.

By (7k) we have I ~ a k - 1 ) .

Let Kk be the polynomial hull of the set

BkUfk(Ak+~UI~).

Since f k ( C ) is a complex submanifold of C ~ and Bk is polynomially convex, it is seen easily that Kk is contained in

BkUfk(C),

and it is the union of

Bkufk(Ak+lUI ~

and the bounded connected components of the complement

fk(C)\(BkUfk(Ak+l UI~

(see L e m m a 5.4 in [7]). Note t h a t (Tk) and ( e ) i m p l y that

fk(R\(--ak+l, ak--1))C

C n \ K k .

Choose R k + l > R k + l such that

KkC(Rk+I-1)B,

and let

Bk+I=Rk+IB.

Choose

ak+l>ak+2

to get (6k+1). We now want to approximate /~ on the larger interval

Ik+l=[-ak+~,a~+~]

by the image of the next embedding Cr n (to be constructed). In order to apply L e m m a 3.4 we first approximate A as follows:

4.1. L e m m a .

There exists a proper C ~ embedding

A~: Rr ~

satisfying

( i ) ) ~ = f ~

on [--ak+l, ak--1],

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A C a r l e m a n t y p e t h e o r e m for p r o p e r h o l o m o r p h i c e m b e d d i n g s 165

(ii)

^A~=A

on R\Ik,

(iii) IA(~)(t)-A(~)(t)l <~(t)

for t~Ik \(-a~ + l, ak--1), O<_s<_r,

(iv) A(~)(t)=A(~)(t)

for t~T, O<_s<_r, and

(v)

A~(t)~Kk when Itl>a~-l.

Proof.

We define the cutoff function Xk on R using X, so that Xk = 0 on R \ I ~ , x k = l on

[--ak+l,ak--1],

and

IIxkhllc~<CIIhllc~

as before. Let

k(t) t

By Lemma 3.1, (3k), (4k), and choice of zl and ~k, we see that (i) (iv) are satisfied for Ak in place of Ak.

To obtain (v) we use a transversality argument to perturb Ak on the set

Ik\(--ak§ ak--1).

First note t h a t if

[t[>ak,

then

]Ak(t)[=[A(t)[>Rk+l

by (6k), so

Ak(t)~Bk.

Also, by (hk), we see t h a t Ak(t)~fk(C), so

Ak(t)q~Kk.

Next, if

teTn(Ik\(--ak+l, ak--1)),

then by (4k), (7k), and (e) we see that

Ak(t)=fk(t)q~

Kk. Hence there exists a neighborhood V of

TN(Ik\(--ak+l,ak--1))

such t h a t J,k (V) n = O.

Thus we need only perturb Ak on

Ik\(VU(--ak+l,ak--1))

to get (v). Note that if

tEIk\(--ak+l, ak--1),

then from (6k) and (2k) we see that

[Ak(t)[>Rk+89

so Ak(t)~Bk. Finally, a simple transversMity argument implies that we can make an arbitrarily small C ~ perturbation of Ak to avoid fk(C), and hence we get Ak with Ak=Ak outside

Ik\(VU(--ak+l,ak--1))

and Ak satisfying (i)-(v). []

Now we can use Lemma 3.4 to approximate Ak, hence to approximate A. Set 5k+1 = min{~](t):

t E Ik+l } /2C,

O'k+ 1 ~--- min{~](t)-[A (~) ( t ) - A (~) (t)]:

t

E I k + l , 0 < 8 < r } > O.

Choose r so small that

~<min{2-(k+l),hk+l,ak+l} , f k ( A j + c k B ) + c B C I n t B j , l <_j<_k.

Apply L e m m a 3.4 with )~=),k,

f = f k , I=[--ak+l, ak--1], K=Kk, a=ak+l, r

and T unchanged, and ~ as above. This provides ~k+l EAut C n and G = ~ k + l

~

EAut C n satisfying

{ l~kq_l(Z)--z[

< ~ for

z e K k ,

hence on

Bk;

for teIk+l, 0 < s < r ;

G (s)

(t) = A~ ) (t) for

t e Tnlk+l, 0 < s < r.

In particular, (2k+1)-(4k+~) hold with G in place of

fk+l.

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Since f k ( A k + l ) c K k C

(Rk+l-

1)B, we can choose r <ok small enough that (lk+1) holds with G in place of

fk+l

and ~+1 in place of r and such that if r C ~ with II~p(t)-tltc~(ik+~) < ~ + 1 , then (2k+l) and (3k+1) hold with Gor in 1 , Then with G in place of fk+l, we have (lk+l)- place of fk+l. Let r ~ck+ 1.

(4k+l), (6k+l), and G ( - a k + l ) ,

G(ak+l)6Bk+l

by (6k+l) and (2k+1).

Next we want to obtain (7k+1). We do this using Lemma 3.2 to change the embedding so that the image of R \ I k + l misses Bk+l while leaving the embedding essentially unchanged on Ak+lt3Ik+l. Apply Lemma 3.2 with

A=G-I(Bk+I), ~=

k + l , / = I k + l , r unchanged,

Z=TNIk+~

and r ~Ek+l. This gives a shear 1 Ck+l(Z) : Z+gk+ l ( z l ) e 2

with _1c

I~k+I(Z)--ZI < ~ k+l, Z C A k + I ; 1C -

I[r < ~ k+l,

gk+l(f;) = O, (s) t E T N I k + I , 0 < S < r;

Ck+l (t) ~ a - l(Bk+l), t E a \ I k + l .

Let

H=GoCk+I.

Then with H in place of fk+l, we have (lk+l)-(4k+l), (6k+1), and (Tk+l).

For the final correction, we use Lemma 3.3 to obtain (5k+l) while maintaining the other properties. Let

R>ak+l

be such that

A=G-I(Bk+I)CRB.

Let 5>0 be such that

Ck+l ( I - R , R ] \ ( - - a k + l , a k + I ) + S B ) N A = 0, and such that if BEAut C ~, with

10(z)-z] <5

on RB, then

(1)

ffCk+~

oSJR(t)-tllc~(xk+,) < ~k+,.

Apply Lemma 3.3 with A replaced by H-loA, K = R B , r unchanged,

Z=TNIk+I,

and r 89162 This gives a shear

8k+l(Z)=z+hk+l(Zl)Vk+l

with 7rlvk+l =0 such that

IOk+I(Z)--Zl

< min{5, 89162

8k+1 (C) AH-1A(R) =

H-1A(TNIk+I);

hk+l(t)

(s) =0,

z E Ak+l;

tETNIk+I,

0 < s < r , and such that (1) holds with O----Ok+ 1. Also, by the choice of R and 5,

~/)k+l

~ (a\fk+l)NA

= O.

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Taking ~k+l=r and

fk+l = HoOk+l = Ok+l o fko~k+l

we obtain (5k+1) and preserve the remaining hypotheses. Hence we obtain ( l k + l ) - (7k+1). Note that ( k + l ) B C B k + l so we also obtain (a)-(f), thus finishing the inductive step.

The case k--1 is similar to the general step. First apply Proposition 2.1 with K = 0 , C = [ - 3 , 3] c C , F : A , ~ : C -1 inf{~(t):re [-3, 3]}, and Z:TFfl [-3, 3] to get r EAut C n satisfying the conclusions of that proposition. Choose R1 >_2 such that r choose a 1 > 4 to get (61), and let I i = [ - a l , a l ] . Choose 51 to satisfy (f) for j----1.

Define a proper C ~ embedding )~0 as in Lemma 4.1 so that (i)-(v) are satisfied with A0 in place of Ak, r in place of fk, 3 in place of he, [--3, 3] in place of Ik, and r (A1) in place of Kk. Apply Lemma 3.4 with )~=)m, f =r I = [-2, 2], K = r (A1), a=al, r and T and r unchanged. This gives r C n such that

Ir < z e

a n d s u c h t h a t O 1 = r 1 6 2 2 1 satisfies

{ IlOl- 011c-u,)<e;

O~8)(t) = ; ~ ) (t), t ~ T n I 1 , 0 < s < r .

As before, we can apply Lemmas 3.2 and 3.3 to obtain e l > 0 and a shear ltI/1 such that the hypotheses (11)-(71) hold for f 1 = O 1 o ~ 1 , and (a)-(f) hold for k = l . This completes the base case.

To finish the proof of Theorem 1.1, note that

k

9 1 ... k(z)=z+Z(g (zl) 2+hj(zl>j)

j = l

and that (c) implies

Igj(zl)e2q-hj(zl)vjl < 2 - j , Izll

<j.

Hence this sum converges uniformly on compacts to a shear ~ ( z ) = z + G ( z l ) for some holomorphic map G: C--+{0} • C ~-1.

By Proposition 4.2 in [7], the composition Ok ... O1 converges locally uniformly to a biholomorphic map from a domain f~ onto C n, and

U ...

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We claim t h a t ~I'(Cx{O})CfL Let k > l . By (lk) we have 9 1 . . . ~ k ( a k - l + ~ k g ) C (~k . . . ~ l ) - l ( B k _ l ) .

O 0

Since I ~ P j ( z ) - z l < c d on A k _ l for j > k , and ~ j = k + l cdKek by (c), we see t h a t l i m ~IJkq_ 1 ... ~I] m (z) E Ak-1 + E k B

for each z E A k _ l . Hence

lI/(/k k_ 1) C ((I) k ... 491)-1(Bk_1) C f~, k > l ,

so the claim holds. In particular, ~oq~: C - - * C n is a proper holomorphic embedding.

Finally, using the conditions (lk)-(7k), we see t h a t ~ o ~ : C~--*C n is a proper holomorphic embedding with the desired properties. []

R e f e r e n c e s

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14. ROSAY, J.-P. and RUDIN, W., Arakclian's approximation theorem, Amer. Math.

Monthly 96 (1989), 432-434.

15. ROSAY, J.-P. and RUDIN, W., Holomorphic maps from C '~ to C '~, Trans. Amer.

Math. Soc. 310 (1988), 47-86.

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J. E., ed.), pp. 563-569. Math. Notes 38, Princeton Univ. Press, Princeton, N. J., 1993.

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Received April 18, 1996 Gregery T. Buzzard

Department of Mathematics Indiana University

Bloomington, IN 47405 U.S.A.

Franc Forstneric

Department of Mathematics University of Wisconsin Madison, WI 53706 U.S.A.